Tutorial 8: The Costs of Production (cont.)

In the last Tutorial we developed and practiced
using the isocost/isoquant model to see how a firm chooses
the cost-minimizing combination of inputs that would get
it the desired level of output. Now we look at the differences
in the cost of increasing output in the long run (when no
inputs are fixed) and in the short run (when at least one
input is fixed).

Cost Minimization in the
Short Run and in the Long Run

The Long-Run Expansion Path

Suppose the firm's production function gives
it constant returns to scale (need a review?).
Figure 8 illustrates the impact on the firm's costs of increasing
its output by one and one-half times from 100 to 150 units/day.
Because the firm faces constant returns to scale, it will
have to hire 1.5 times its original number of inputs, so
L and K increase from (5, 5) to (7.5, 7.5). As a result,
the cost of producing the new output to increases by 1.5
times, from $20 to $30.

The two cost-minimizing combinations of labor
and capital, and any number of other such combinations we
could generate, can be plotted together as a line from the
origin showing all the cost minimizing input combinations
for various levels of output this firm could produce in
the long run. It is referred to as the firm's long-run
expansion path (LREP). It shows the cheapest cost of
raising output in the long run.

Figure 8

Short-Run Expansion Path

Suppose the firm could not expand the use
of both inputs, i.e., suppose the firm is looking
to expand output in the short run? Let's freeze K at 5 hours
per day. Figure 9 shows the result. To increase its output
from 100 to 150 units per day, given its production function,
the firm will have to increase the amount of labor from
5 to 11.25 hours per day (K = 5 hours/day). This raises
total cost from $20 to $32.50. Thus, when the firm expands
output in the short run it faces higher costs. This is due
to the inflexibility of the production process in the short
run when at least one input is fixed. The short-run expansion
path (SREP) illustrates the minimum cost of increasing
output in the short run. Because the isocost line C1
lies above the point of tangency with isoquant Q1 ,
we can see that even this minimum short run cost is more
expensive than the minimum cost possible in the long run.

Figure 9

Long-Run Average Cost

Economies and Diseconomies of Scale

When a firm's production function gives it constant returns
to scale, as does the one pictured in Figure 9, a doubling
of both inputs will double output. Well it will also double
cost. So average cost, C(Q)/Q, will remain unchanged. Let's
see that mathemagically:

AC1(Q) = C(Q)/Q
AC2(Q) = 2 · C(Q)/2 · Q\ AC2(Q) = AC1(Q)

But what about the other types of returns to scale? When
a firm's production function gives it increasing returns
to scale a doubling of both inputs will more than double
output. Of course it will also double cost. So average cost
will fall.

AC1(Q) = C(Q)/Q
AC2(Q) = 2 · C(Q)/3 · Q\ AC2(Q) = (2/3) ·
AC1(Q)

Thus a firm enjoying increasing returns to
scale is likely to experience economies of scale
when it increases its output in the long run. With economies
of scale, a firm can double output at less than double the
cost.

Similarly, when a firm's production function
gives it decreasing returns to scale a doubling of
both inputs will less than double output. Of course it will
also double cost. So average cost will rise.

AC1(Q) = C(Q)/Q
AC2(Q) = 2 · C(Q)/1.5 · Q\ AC2(Q) = (1 1/3)
· AC1(Q)

Thus a firm has decreasing returns to scale
is likely to experience diseconomies of scale when
it increases its output in the long run. With diseconomies
of scale, a firm can double output but at more than double
the cost.

Another way to look at how much more expensive it is to
increase output in the short than in the long run, is to
look at a graphical model comparing short run and long run
average costs. Figures 7.8 and 7.9 (p. 225) in P&R provides
two specific illustrations. In the short run, the firm is
moving along its short run average cost curve (like the
one you generated for Tutorial 8 Question 8 in the CostMin.xls
workbook). In the long run it moves along a long-run average
cost curve. The long-run average cost curve is an envelope
function (don't worry about it) that is just tangent to
a set of short run average cost curves, each of which represents
a short run increase in the scale of production. The shape
of the long-run average cost curve depends on whether the
production function generates increasing, constant, or decreasing
returns to scale.

In part d we used the isocost/isoquant model
to discover the differences in the cost of increasing output
in the short run (when at least one input is fixed) and
in the long run (when no inputs are fixed). We found that
increasing production in the short run was much more expensive
that increasing output in the long run. This was the result
of the inflexibility of the production process in the short
run (when at least one input is fixed).

Now it's time to look at the revenue side
of production. Then we will be in the position to determine
how a profit-maximizing firm chooses how much output to
produce.